Zentrale Ideen der numerischen Mathematik - Vorschlag eines Katalogs und unterrichtliche Umsetzungen
Titz, Mauricio Marvin; Heitzer, Johanna (Thesis advisor); Humenberger, Hans (Thesis advisor)
Dissertation / PhD Thesis
Dealing with numbers and calculating results are elementary components of mathematics and have always been intricately connected with their practical usability. As far back as the ancient Egyptians, mathematical calculations were even used to build pyramids. Over the course of the subsequent centuries, mathematics has been continuously developed as a structural science on the one hand, but also as a discipline with ever more far-reaching applications on the other. The significance of mathematical calculations for technical progress has steadily increased over the past decades. Today, the prompt availability of large amounts of data and powerful calculating machines enable ever more extensive applications, yet at the same time lead to an increasing demand for suitable mathematical methods. One such example is computer tomography: In addition to technical progress, the further development of mathematical methods has been and continues to be a decisive component in calculating sectional images from a large quantity of measured values and to thus literally be able to look inside bodies. Similar instances of the contribution of mathematics to technical progress can be observed wherever any calculations have to be performed. All kinds of computer-aided simulations and designs in the engineering sciences, but also activities in newer fields such as Big Data or Data Science are hardly conceivable without highly developed numerical calculation methods. Nevertheless, despite their increasing significance for various application disciplines, numerical calculations play a subordinate role in (secondary) mathematics lessons. This is ambivalently intensified by the increasing use of digital mathematical tools to solve application tasks. On the one hand, these tools often function numerically and thus lead to the direct use of numerical methods, on the other hand, it is precisely through them that the challenge of mathematical calculations can be specifically avoided and even obscured. The use of digital mathematical tools is only rarely used as an opportunity to address the calculation methods with their specific problems and properties, thus making the calculation methods themselves the subject of instruction. This work is motivated by this contrast: The intention is to show the potential of the mathematical field of numerics for school lessons and make it more accessible for both secondary levels. For this purpose, numerics is first dealt with from a didactic perspective in order to extract important basic ideas. Based on this, we want to present possibilities of how aspects of numerical mathematics can be implemented in school. The following products are the result: A catalog of central ideas of numerics: Using a catalog of ideas, basic concepts are described and their relevance for the technical discipline, research, university and school mathematics is shown. On the basis of a didactic analysis, the significance of these concepts is justified using selected criteria. As a final step, the catalog is considered in its entirety with regard to possible deficits and redundancies in order to present a coherent construct. An essential feature of the ideas worked out is the possibility to concretize them at school level as well as at current research level. Thus, this catalog can help to develop a better idea of what numerical mathematics is and what characterizes it. Teaching materials and teaching suggestions: Teaching materials are presented for the central ideas presented in the catalog, which have been tested with groups of pupils, discussed with active teachers and further developed on this basis. In addition to smaller teaching suggestions, which are intended to serve as inspiration for interested teachers, two more comprehensive workshops are presented: Using computed tomography as an application context, high school students work on essential questions of numerical mathematics, such as how to solve linear systems of equations. Besides the manual execution of various computational methods, the workshop also focuses on investigations with dynamic worksheets (GeoGebra) and prepared working environments with special software (Matlab and GNU Octave). Dealing with inaccurate values and calculation results will be the focus of another workshop, the materials of which can be used from the intermediate level onwards. In addition to raising awareness for a reasonable degree of accuracy, we will discuss application contexts in which the exact calculation of results is neither possible nor reasonable. Two insights are tangible on the basis of these products: On the one hand, an increased integration of numerical basic concepts into school lessons is both possible and profitable and meets with the interest of both teachers as well as students. Secondly, it will be demonstrated by way of example of what materials can look like for implementation in teaching, based on a sustainable didactic concept. In summary: Numerical mathematics can be used to make school lessons practical and authentic. On the one hand, it offers the potential to discuss application- and research-related issues with students, which is conducive to the construction of a realistic conception of mathematics. On the other hand, the sound elementary nature of the contents makes it possible to adapt the level of requirements to different age groups, so that competence development can be implemented across the different age groups right up to university mathematics.